Arithmetic instructs the algorist on the left and the abacist on the right. This illustration attempts to depict a conflict between calculation and counting mathematics.
This site is not intended to be a substantive historical work on the subject of the history of mathematics, but I felt compelled to include a little background so that we can appreciate the depth and breath of the history of math. With this history in mind, we can see that the scientific calculator represented a, glorious, fleeting moment in the progression of man's understanding of science and engineering. I've arbitrarily classified the basic components of mathematics into five broad areas:
This arbitrary breakdown also gives us some insight into how functions on modern scientific calculators are segmented. Each of these areas requires specific computational resources and it is these resources that make the design of good scientific calculators a challenge.
Before launching into any detail, I thought that it would be a good idea to discuss some mathematical concepts that we all know a little about, but may have forgotten. The foundations of some the topics I'll discuss later go back as far as 5000 B.C. but mathematics as rigorous intellectual, abstract, pursuit didn't get interesting until about 300 B.C. This was the rough time frame of the publication of Euclid's The Elements. By any standard, The Elements is one of the most enduring books in history. Over 1000 editions have been published in basic form right through the twentieth century. Nothing in Elements was written in stone. As a matter of fact it was written on parchment and rolled onto wooden sticks. Longer works were cut into shorter strips and then collected as a "book". One of the most unique things about Elements is that it exists at all. Much of the work from this time disappeared because of fragile nature of the media. I often wonder what, from our era, individuals will be reading in the year 3000. The Greeks and Euclid in particular left us with a solid basis for the development of later mathematical thought, but they also saddled us with a lot of baggage, which would have to be shed before our understanding of mathematics began to grow. The Greeks tended to look at numerology in a spiritual way and often searched for deep meanings in the nature of mathematics. Euclid played a great role in recording historical facts as well as discovering some interesting mathematical forms. Euclid is often associated with geometry, but we wouldn't be giving the guy enough credit if we didn't also remember him for his work in the theory of numbers. Euclid did much of the founding work in the theory of prime numbers and "perfect" numbers. Both areas are still fertile areas for discussion today (well, for those of us with no social lives to speak of.....) At the risk of adversely affecting your social life, let's take a look at perfect and prime numbers.
A Perfect number is defined as a number, the sum of its aliquot divisors ( or factors ) equals the number itself; by aliquot, we mean all the factors of the number including unity (1), but not the number itself. The number six (6) is the smallest perfect number: its factors(not counting itself) are 1,2, and 3. Their sum is six. The next perfect numbers are 28, 496, 8128, and 3,355,036. Can you see the pattern? There isn't one, but Euclid deduced that a perfect number could be always be represented by the following algebraic form: (2^(n - 1))X(2^n-1) , where ^ represent "to the power of " and n is a positive integer such that 2^n -1 is prime. The first five such solutions to this are n= 2, 3,5,7, and 13. These numbers all happen to be prime. This might lead one to assume that 2^prime - 1 = a prime number, but Euclid also knew that couldn't be proven. In fact it has been shown that there are prime numbers that do not result in a new prime number when used in that equation. Euclid simultaneously introduced broad theoretical concepts to numerology, while insisting upon rigor and logic to confirm the concept as well as the result. It's important to remember that this was happening over two thousand years ago: raising 2 to an integer power (that also happened to be prime), building a series of numbers whose magnitude rapidly reached into the millions . To this day, mathematicians still can't predict the occurrence of a perfect number between two arbitrarily spaced primes! Another interesting point, there was no "algebra" during this period and the Greek arithmetic was not ten based.
A golden rectangle was defined by the Greeks as a rectangle whose sides "a" and "b" obey the following mathematical convention: a/b = (a+b)/a . This is a quadratic equation that has a solution that can be found by setting b= 1 and solving the equation such that a = (a+1)/a . The solution to the quadratic is (1 + sqrt(5))/2 or about 1.618. The Greeks considered this ratio to be the most visually pleasing ratio.
Mathematics is often thought of as a genteel, scholarly pursuit. For the most part, the history of mathematics reflects the history of the civilizations from which it grew. In this section, we'll take a look at people who were killed as a direct consequence of their interaction with math or science.
Archimedes gets the shaft.....literally
Probably the best known mathematician to have died a violent death while practicing math was Archimedes. Not much detail is known about his life, but his work and his death have been well chronicled. Archimedes lived in Syracuse. Between 214 and 212 B.C. (The Second Punic War) the city was besieged by Roman solders. Archimedes is known to have helped the battle against Rome through his development of unique weapons such as catapults, pulleys, and devices to set fire to ships. The account of Archimedes death comes from the account of Plutarch's description of the life of the Roman General, Marcellus. According to this account, Marcellus ordered that Archimedes be spared during the sack of Syracuse. It is said that a soldier came upon Archimedes as he drew a diagram in the sand. The soldier ordered him to move and Archimedes waved him off. The Roman soldier killed him on the spot. While the Romans were never known to be easy on the conquered populace, they were really hard on the hired help. Marcellus ordered the soldier and his entire family killed.
Hypatia's opinions are somewhat unappreciated.....
Technical education of women (or any education for that matter) is a relatively new concept. The story of Hypatia of Alexandria underscores the contributions that were probably lost as a result of the practice of not educating women. The spirituality of the Greeks has been well documented but what isn't well known is degree to which mathematics was combined with the mystical aspects of Greek life. Mathematics was woven in the fabric of Greek mysticism. The search for perfect numbers was more than a curiosity, it was a religious imperative. Out of this back drop came Hypatia. She was the daughter of Theon, the mathematician. Both she and her father were pagans, Platonists and Pythagoreans. She was a superb scholar and she was beautiful. It is said that she lectured from behind a screen to save the students from distraction (as opposed to today, where teachers have to lecture from behind bullet proof glass to save their lives). She lived during the third century AD and the politics of religion had begun its assault on the intellectual process. The absolutism of the Christian church, coupled with the expectation of the Second coming of Christ, made intellectual pursuits a waste of time. Who needed intellect if Christ was coming back? (Unlike today, where the government has replaced the second coming with social programs...) Hypatia didn't care much for the Christians and she made it known. To make matters worse she got involved in local politics. She also had a bad habit of stopping strangers and lecturing them on the philosophies that she espoused (hey, she was a woman, you know). She ended up getting caught in a feud between two politicians (one a former student and the other, the Patriciarch of Constantinople ). The Christians fell upon her during an anti-Roman demonstration. They dragged her into a church, stripped and murdered her and then quartered and burned her corpse...all in devotion to Christ (At least they didn't keep their feelings inside....) . Hypatia was the last of the pagan scientists. Her death in 415 A.D. coincided with the acknowledged end of the Roman empire and predicated the Dark Ages. No substantive advances in the knowledge of nature or math would occur in Christian Europe for the next 1000 years. Hypatia's untimely demise was to become a call to arms for generations of European free-thinkers, scientists , and anti-Catholics in a battle for intellectual freedom that rages to this day.
Maybe we should start with the question: What is a sexagismal number? We deal with sexagismal numbers all the time ( yes, that was a very bad pun). sexagismal numbers are based upon fractions of 60. Sexigismal numbers are the bases of our measures of degrees , minutes, seconds and hours, minutes, and seconds. Now we have a new question: How did the division of a circle and the division of a day come to have the same fractional components? The answer lies in a number system based on the sum of fractions of 60 that preceded both the notions of length of a day and division of the circle. As you might guess from that statement, sexagismal numbers are old; very old.
Greek astronomy held that the daily passage of the sun through the heavens relative to the fixed stars was a constant angular distance. The passage took approximately 360 days (hey, who's counting?) and the concept of the degree was born. It's interesting to note that we have lived with a system of measure based upon an incorrect estimate of the length of a year and a model of planetary motion that was just plain wrong.
To get to the first vestiges of a base 60 numeric system we have to go back almost 6000 yes to the region then known as Sumeria. Sumerians were the first city dwellers. They actively traded goods, owned land, and by all accounts, had their share of disputes about price, ownership and value. In short: same problems, different millennia. The trade issues necessitate the development of a written language, and numerical system. They weren't worried about recording history, they were interested in recording profit. The mathematics of Sumeria preceded the invention of an alphabet by a few millennia, so it took a bit of time to decipher what they were recording and the numerology had a profound defect, which I'll talk about in a moment. Like the modern Arabic notation we use, the Sumerian numerical system was ordered right to left. For comparison purposes, let's take a look at our system first. The number 349 is really a short form of the numeric expression (3*100) + (4*10) + (1*9) + (0). The Sumerian expression 3,4,9 could mean (3*60) + (4*1) + (9/60) or it might mean (3*1) + (4/60) + (9/360). The numerology of Sumeria was context based, they didn't have a position for zero. They could form numbers with extreme precision, but without a context, the magnitude remains unknown.
The Sumerians recorded their numbers on clay tablets using a stylus. Tables were calculated of fractions of 60 and numerical equivalents were derived for sums of fractions. The clay dried and the records, while somewhat hard to decipher, represented a monumental amount of accumulated calculation. This effort was, as they say, "cast in stone". Over 4 millennia, Sumeria came and went, but those hardened tablets were recorded by many later civilizations .The beauty of the Sumerian system lay in it's extreme precision in fractions. The factors of 60 are 1,2,3,4,5,6,10,12, 15, 20 and 30. In the base 10 system the factors are 1,2, and 5. There is no exact numerical representation for the value of 1/3 in a base 10 system but in Sumerian notation it is simply 20/60. The paradox of Sumeria is the extreme precision coupled with a total lack of accuracy. This a common mathematical thread which holds older civilizations together. The fractions that the Sumerians labor sly cast in stone would become the basis for calculation of astronomical movements 4000 years later by the Greeks, and used in timing calculations to land a man on the moon over 6 millennia later.